Taking the long way around

The Chicago Art Museum's Renaissance display consists of four hallways bordered around a square courtyard. A single guard is assigned to patrol the four hallways. When the guard starts working, he begins in one of the corners and walks clockwise. When he arrives at a subsequent corner he flips two coins. If both coins are heads, he changes the direction he is walking. Otherwise, he continues in the same direction. Let \(E\) be the expected number of lengths of hallway that he walks before he first returns to his starting corner. Let \(p\) be the probability that he walks strictly more than \(E\) lengths of hallway before returning to his starting corner. \(p\) can be expressed as \(\frac{a}{b}\) where \(a\) and \(b\) are coprime positive integers. What is the value of \(a + b\)?

Details and assumptions

The guard may walk in the same hallway more than one time. Each time he walks in it counts as one length of hallway.

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